2019 AIME I Problems/Problem 5
Problem 5
A moving particle starts at the point and moves until it hits one of the coordinate axes for the first time. When the particle is at the point
, it moves at random to one of the points
,
, or
, each with probability
, independently of its previous moves. The probability that it will hit the coordinate axes at
is
, where
and
are positive integers. Find
.
Solution
A move from to
is labeled as down (
), from
to
is labeled as left (
), and from
to
is labeled as slant (
). To arrive at
without arriving at an axis first, the particle must first go to
then do a slant move. The particle can arrive at
through any permutation of the following 4 different cases:
,
,
, and
.
There is only permutation of
. Including the last move, there are
possible moves, making the probability of this move
.
There are permutations of
, as the ordering of the two slants do not matter. There are
possible moves, making the probability of this move
.
There are permutations of
, as the ordering of the two downs and two lefts do not matter. There are
possible moves, making the probability of this move
.
There are permutations of
, as the ordering of the three downs and three lefts do not matter. There are
possible moves, making the probability of this move
.
Adding these, the total probability is . Therefore, the answer is
.
Solution by Zaxter22
See Also
2019 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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